A horizontal uniform thin rod AB of mass m and length $l_{0}$ rotates freely about a stationary vertical axis $O O^{'}$ passing through the end A, which is located exactly midway between $O a n d O' a n d O O' = l$ . At what angular velocity (in SI units) of the rod is the horizontal component of the force acting on the lower end of the axis $O O'$ equal to zero. $(g = 10 m / s^{2}, l = 5 m, l_{0} = 3 m)$

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Detailed Solution

About O

$N' l + τ_{c e n t r i f u g a l} + (- m g \frac{l_{0}}{2}) = 0$

About $O'$
$N l + τ_{c e n t r i f u g a l} + (- m g \frac{l_{0}}{2}) = 0$
$τ_{c e n t r i f u g a l} = \int_{0}^{l} \frac{m ω^{2} l}{2 l_{0}} x d x = \frac{1}{4} m ω^{2} {ll}_{0}$ (anticlockwise)
$∴ N' l + \frac{1}{4} m ω^{2} {ll}_{0} - m g \frac{l_{0}}{2} = 0$
and $N l - \frac{1}{4} m ω^{2} {ll}_{0} - m g \frac{l_{0}}{2} = 0$
When $N' = 0, \frac{1}{4} m ω^{2} {ll}_{0} - m g \frac{l_{0}}{2} = 0$